Some new q-congruences on double sums

被引:17
作者
Wang, Xiaoxia [1 ]
Yu, Menglin [1 ]
机构
[1] Shanghai Univ, Dept Appl Math, Shanghai 200444, Peoples R China
关键词
q-Congruence; Cyclotomic polynomial; Basic hypergeometric series; Supercongruence; SUPERCONGRUENCE;
D O I
10.1007/s13398-020-00946-9
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Swisher confirmed an interesting congruence: for any odd prime p, Sigma((p-1)/2)(k=0) (-1)(k) (6k + 1) (1/2) 3k/k!(3)8(k) Sigma(k)(j=1) (1/(2 j - 1)(2) - 1/16 j(2)) = 0 (mod p), which was conjectured by Long. Recently, its q-analogue was proved by Gu and Guo. Inspired by their work, we obtain a new similar q-congruence modulo Phi(n)(q) and two q-supercongruences modulo Phi(n)(q)(2) on double basic hypergeometric sums, where Phi(n)(q) is the n-th cyclotomic polynomial.
引用
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页数:10
相关论文
共 22 条
[1]  
Gasper G., 2004, ENCY MATH ITS APPL, Vsecond
[2]   q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon [J].
Gorodetsky, Ofir .
INTERNATIONAL JOURNAL OF NUMBER THEORY, 2019, 15 (09) :1919-1968
[3]   A q-ANALOGUE OF A HYPERGEOMETRIC CONGRUENCE [J].
Gu, Cheng-Yang ;
Guo, Victor J. W. .
BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 2020, 101 (02) :294-298
[4]   q-Supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping [J].
Guo, Victor J. W. .
ADVANCES IN APPLIED MATHEMATICS, 2020, 120
[5]   Proof of Some q-Supercongruences Modulo the Fourth Power of a Cyclotomic Polynomial [J].
Guo, Victor J. W. .
RESULTS IN MATHEMATICS, 2020, 75 (03)
[6]  
Guo VJW, 2020, RACSAM REV R ACAD A, V114, DOI 10.1007/s13398-020-00854-y
[7]   Proof of a basic hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial [J].
Guo, Victor J. W. ;
Schlosser, Michael J. .
JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, 2019, 25 (07) :921-929
[8]  
Guo VJW, 2020, RAMANUJAN J, V52, P123, DOI 10.1007/s11139-018-0096-6
[9]   A q-microscope for supercongruences [J].
Guo, Victor J. W. ;
Zudilin, Wadim .
ADVANCES IN MATHEMATICS, 2019, 346 :329-358
[10]   A q-analogue of the (L.2) supercongruence of Van Hamme [J].
Guo, Victor J. W. .
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2018, 466 (01) :749-761